Let $X$ be the set of all continuous and real-valued functions on $\mathbb{R}$. X is a commutative ring with pointwise addition and multiplication. Let $\alpha \in \mathbb{R}$ be arbitrary.
(1) Show that $I = \left\{f \in X \ \big\vert \ f(\alpha) = 0\right\}$ is a maximal ideal of $X$.
(2) Show that $X/I$ is isomorphic to $\mathbb{R}$
(1)
I tried to visualize this task. The only reason $I \neq X$ is position $\alpha$, where I cannot create another value than 0 using translation or rotation. If I take any function $f'$ for which $f'(\alpha) \neq 0$ applies, I can use the fact that $X$ contains all functions except for the ones being not 0 at $\alpha$, and can therefore create any continuous, real-valued function on $\mathbb{R}$.
While this seems clear visualized at the paper, I have no clue how to formally write this down, can you please help me to formalize it? I'm especially stuck because I don't know how to formally write down all continuous, real-valued functions on $\mathbb{R}$.
(2)
I need to find a bijection between $X/I$ and $\mathbb{R}$, but don't understand why this is possible. As far as I understand $X/I$ are all continuous real-valued functions on $\mathbb{R}$ that are not $0$ at position $\alpha$. But I can find more than one of those functions for each $r \in \mathbb{R}$ which is in $X/I$. So how could I find a bijection? Where is my mistake?
(2) implies (1), for an ideal is maximal iff its quotient ring is a field. As for (2), the map $e_\alpha \colon X \to \mathbb{R}$, $e_\alpha(f) = f(\alpha)$, is a ring homomorphism with kernel $I$.